Physics Department, Yarmouk University, Irbid Jordan

Phys. 201 Mathematical Physics 1

Dr. Nidal M. Ershaidat Doc. 12

Author’s email: enidal@yu.edu.jo, Address: Physics Department, Yarmouk University 21163 Irbid Jordan

1

Fourier Series

Definition

A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions. The computation and study of Fourier series is known as harmonic analysis and is extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical.

Illustration

Examples of successive approximations to common functions using Fourier series are illustrated below.

Application in the solution of ordinary differential equations

In particular, since the superposition principle holds for solutions of a linear homogeneous ordinary differential equation, if such an equation can be solved in the case of a single sinusoid, the solution for an arbitrary function is immediately available by expressing the original function as a Fourier series and then plugging in the solution for each sinusoidal component. In some special cases where the Fourier series can be summed in closed form, this technique can even yield analytic solutions.

Generalized Fourier Series

Any set of functions that form a complete orthogonal system have a corresponding generalized Fourier series analogous to the Fourier series. For example, using orthogonality of the roots of a Bessel function of the first kind gives a so-called Fourier-Bessel series.

2

Computation of Fourier series

The computation of the (usual) Fourier series is based on the following integral identities which represent the orthogonality relationships of the sine and the cosine functions:

(((( )))) (((( ))))∫∫∫∫ππππ++++

ππππ−−−−

δδδδππππ==== nmdxxnxm sinsin (1)

(((( )))) (((( ))))∫∫∫∫ππππ++++

ππππ−−−−

δδδδππππ==== nmdxxnxm coscos (2)

(((( )))) (((( ))))∫∫∫∫ππππ++++

ππππ−−−−

==== 0dxxnxm cossin (3)

(((( ))))∫∫∫∫ππππ++++

ππππ−−−−

==== 0dxxmsin (4)

(((( ))))∫∫∫∫ππππ++++

ππππ−−−−

==== 0dxxmcos (5)

For 0≠≠≠≠nm , , where δmn is the Kronecker delta. Using the method for a generalized Fourier series, the usual Fourier series involving sines and cosines is obtained by taking f1(x) = cos x and f2(x) = sin x. Since these functions form a complete orthogonal system over [- π, π], the Fourier series of a function f(x) is given by

(((( )))) (((( )))) (((( ))))∑∑∑∑∑∑∑∑∞∞∞∞

====

∞∞∞∞

====

++++++++====11

021

n

n

n

n xnbxnaaxf sincos (6)

a0, a1 …an are called the Fourier coefficients. These coefficients are obtained using the following relations:

(((( ))))∫∫∫∫ππππ++++

ππππ−−−−ππππ

==== dxxfa1

0 (7)

(((( )))) (((( ))))∫∫∫∫ππππ++++

ππππ−−−−ππππ

==== dxxnxfan cos1

(8)

(((( )))) (((( ))))∫∫∫∫ππππ++++

ππππ−−−−ππππ

==== dxxnxfbn sin1

(9)

n = 1, 2, 3, …. Note that the coefficient of the constant term a0 has been written in a special form compared to the general form for a generalized Fourier series in order to preserve symmetry with the definitions of an and bn. A Fourier series converges to the function f (equal to the original function at points of continuity or to the average of the two limits at points of discontinuity)

3

(((( )))) (((( ))))

(((( )))) (((( ))))

ππππππππ−−−−====

++++

ππππ<<<<<<<<ππππ−−−−

++++

====

−−−−++++

++++−−−−

ππππ→→→→ππππ→→→→

→→→→→→→→

,limlim

limlim

0

0

21

21

00

xforxfxf

xforxfxf

f

xx

xxxx (10)

if the function satisfies so-called Dirichlet conditions.

As a result, near points of discontinuity, a "ringing" known as the Gibbs phenomenon, illustrated above, can occur. For a function f(x) periodic on an interval [- L, L] instead of [- π, π], a simple change of variables can be used to transform the interval of integration from [- π, π] to [- L, L]. Let

xL

x ′′′′ππππ==== (11)

xdL

dx ′′′′ππππ==== (12)

Solving for x ′′′′ gives xL

xππππ

====′′′′ , and plugging this in gives

(((( )))) ∑∑∑∑∑∑∑∑∞∞∞∞

====

∞∞∞∞

====

′′′′ππππ++++

′′′′ππππ++++====′′′′

11

021

n

n

n

n xL

nbx

Ln

aaxf sincos

(13)

Therefore,

(((( ))))∫∫∫∫++++

−−−−

′′′′′′′′====L

L

xdxfL

a1

0 (14)

(((( ))))∫∫∫∫++++

−−−−

′′′′

′′′′ππππ′′′′====

L

L

n xdxL

nxf

La cos

1 (15)

(((( ))))∫∫∫∫++++

−−−−

′′′′

′′′′ππππ′′′′====

L

L

n xdxL

nxf

Lb sin

1 (16)

Similarly, the function is instead defined on the interval [0,2L], the above equations simply become

(((( ))))∫∫∫∫ ′′′′′′′′====

L

xdxfL

a

2

0

01

(17)

4

(((( ))))∫∫∫∫ ′′′′

′′′′ππππ′′′′====

L

n xdxL

nxf

La

2

0

1cos (18)

(((( ))))∫∫∫∫ ′′′′

′′′′ππππ′′′′====

L

n xdxL

nxf

Lb

2

0

1sin (19)

In fact, for f(x) periodic with period 2 L, any interval [x0, x0+2L ] can be used, with the choice being one of convenience or personal preference (Arfken 1985, p. 769). The coefficients for Fourier series expansions of a few common functions are given in Beyer (1987, pp. 411-412) and Byerly (1959, p. 51). One of the most common functions usually analyzed by this technique is the square wave. The Fourier series for a few common functions are summarized in the table below. The coefficients for Fourier series expansions of a few common functions are given in Beyer (1987, pp. 411-412) and Byerly (1959, p. 51). One of the most common functions usually analyzed by this technique is the square wave. The Fourier series for a few common functions are summarized in the table below.

Examples

Function f(x) Fourier series

sawtooth wave L

x2

∑∑∑∑∞∞∞∞

====

ππππ−−−−

1

sin1

21

n

xL

nn

square wave 112 −−−−

−−−−−−−−

Lx

HLx

H ∑∑∑∑∞∞∞∞

====

ππππ

ππππK,5,3,1

sin14

n

xL

nn

triangle wave T(x) (((( ))))∑∑∑∑

∞∞∞∞

====

−−−−

ππππ−−−−

ππππ K,5,3,12

21

2 sin18

n

n

xL

nn

If a function is even so that f(x) = f(- x), then f(x) sin(nx) is odd. (This follows since sin(nx) is odd and an even function times an odd function is an odd function.) Therefore, bn=0 for all n. Similarly, if a function is odd so that f(x) = - f(-x), then f(x) cos(nx) is odd. (This follows since cos(nx) is even and an even function times an odd function is an odd function.) Therefore, an=0 for all n.

Complex Coefficients

The notion of a Fourier series can also be extended to complex coefficients. Consider a real-valued function f(x). Write

(((( )))) ∑∑∑∑∞∞∞∞

====

====0n

xnin eAxf (20)

Now examine

(((( )))) ∫∫∫∫ ∑∑∑∑∫∫∫∫ππππ++++

ππππ−−−−

−−−−∞∞∞∞

∞∞∞∞−−−−====

ππππ++++

ππππ−−−−

−−−−

==== dxeeAdxexf xmi

n

xnin

xmi (21)

(((( ))))∫∫∫∫∑∑∑∑ππππ++++

ππππ−−−−

−−−−∞∞∞∞

∞∞∞∞−−−−====

==== dxeA xmni

n

n (22)

(((( )))) (((( ))))[[[[ ]]]]∫∫∫∫∑∑∑∑ππππ++++

ππππ−−−−

∞∞∞∞

∞∞∞∞−−−−====

−−−−++++−−−−==== dxmnimnAn

n sincos (23)

5

nm

n

nA δδδδππππ==== ∑∑∑∑∞∞∞∞

∞∞∞∞−−−−====

2 (24)

mAππππ==== 2 , (25) So

(((( ))))∫∫∫∫ππππ++++

ππππ−−−−

−−−−

ππππ==== dxexfA xni

n 21

(26)

The coefficients can be expressed in terms of those in the Fourier series

(((( )))) (((( )))) (((( ))))[[[[ ]]]]∫∫∫∫ππππ++++

ππππ−−−−

++++ππππ

==== dxxnixnxfAn sincos21

(27)

(((( )))) (((( )))) (((( ))))[[[[ ]]]]

(((( ))))

(((( )))) (((( )))) (((( ))))[[[[ ]]]]

>>>>−−−−ππππ

====ππππ

<<<<++++ππππ

====

∫∫∫∫

∫∫∫∫

∫∫∫∫

ππππ++++

ππππ−−−−

ππππ++++

ππππ−−−−

ππππ++++

ππππ−−−−

0sincos21

021

0sincos21

ndxxnixnxf

ndxxf

ndxxnixnxf

(28)

(((( ))))

(((( ))))

<<<<−−−−ππππ

<<<<ππππ

<<<<++++ππππ

====

021

021

021

0

nforbia

nfora

nforbia

nn

nn

(29)

For a function periodic in [- L/2, L/2], these become

(((( )))) (((( ))))∑∑∑∑∞∞∞∞

∞∞∞∞−−−−====

ππππ====n

Lxnin eAxf 2 (30)

(((( )))) (((( ))))∫∫∫∫−−−−

ππππ−−−−====2

2

21L

L

Lxnin dxexf

LA (31)

These equations are the basis for the extremely important Fourier transform, which is obtained by transforming An from a discrete variable to a continuous one as the length L → ∞.

6

References

1. Arfken, G. "Fourier Series." Ch. 14 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 760-793, 1985.

2. Askey, R. and Haimo, D. T. "Similarities between Fourier and Power Series." Amer. Math. Monthly 103, 297-304, 1996.

3. Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987.

4. Brown, J. W. and Churchill, R. V. Fourier Series and Boundary Value Problems, 5th ed. New York: McGraw-Hill, 1993.

5. Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, 1959.

6. Carslaw, H. S. Introduction to the Theory of Fourier's Series and Integrals, 3rd ed., rev. and enl. New York: Dover, 1950.

7. Davis, H. F. Fourier Series and Orthogonal Functions. New York: Dover, 1963. 8. Dym, H. and McKean, H. P. Fourier Series and Integrals. New York: Academic

Press, 1972. 9. Folland, G. B. Fourier Analysis and Its Applications. Pacific Grove, CA:

Brooks/Cole, 1992. 10. Groemer, H. Geometric Applications of Fourier Series and Spherical

Harmonics. New York: Cambridge University Press, 1996. 11. Körner, T. W. Fourier Analysis. Cambridge, England: Cambridge University

Press, 1988. 12. Körner, T. W. Exercises for Fourier Analysis. New York: Cambridge University

Press, 1993. 13. Krantz, S. G. "Fourier Series." §15.1 in Handbook of Complex Variables.

Boston, MA: Birkhäuser, pp. 195-202, 1999. 14. Lighthill, M. J. Introduction to Fourier Analysis and Generalised Functions.

Cambridge, England: Cambridge University Press, 1958. 15. Morrison, N. Introduction to Fourier Analysis. New York: Wiley, 1994. 16. Sansone, G. "Expansions in Fourier Series." Ch. 2 in Orthogonal Functions,

rev. English ed. New York: Dover, pp. 39-168, 1991. 17. Weisstein, E. W. "Books about Fourier Transforms." http://www.ericweisstein.com/encyclopedias/books/FourierTransforms.html. 18. Whittaker, E. T. and Robinson, G. "Practical Fourier Analysis." Ch. 10 in The

Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 260-284, 1967.

Source internet

Weisstein, Eric W. "Fourier Series." From MathWorld - A Wolfram Web Resource. http://mathworld.wolfram.com/FourierSeries.html

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